Well, lowkey would say the best way to improve the paladin is to make them ammunition for catapults...
If you want my opinion, I am not a fan of making the paladin a "bad guy". I know why they did, because everyone and their grandma's dog seems to want an anti-paladin ever since the actual anti-paladin came out in AD&D. In the current edition, the Oath of Vengeance doesn't sit well with me personally. How it feels to me is that they basically are saying, "We know paladins are holy warriors, but hey, don't worry about role playing the devout holy aspect, but keep the powerful mechanical bits." I.e., they really seem to imply that you can skip all the devout stuff if you want. IMO, the role playing parts were there to help mitigate the class. You got all these powers, but the drawback is that you had to actually play your PC like an extremely devout follower, almost like a fanatical. After all, why would a deity give all these extra powers to a person unless they pushed that deity's agenda. Again, that's just my personal tastes though.
That being said, I know removing that will not be a solution for most people. So....
What I would change is getting rid of divine smite. Or at the very least limiting it to demons/devils/undead only. By allowing it to everything, that can cause paladins to overshadow fighters, especially if you allow frequent rests. I would probably make it more like the favored enemy thing of the ranger. Upon creation, depending on theme, choose one type of of monster, and you can divine smite them. What I see all to often is that paladins don't even cast spells anymore--they just save it for divine smites. And as a player of a cleric in our biggest campaign, it would be nice from a party contribution stand point if the paladin also used spells to help the party, especially when I was low, rather than "save them" for divine smite.